In[]:=
SetDirectory[NotebookDirectory[]];<<ABCInterpolator.m​​
In[]:=
X1=Collect[n^(-p)Integrate[(t^(5/2)/n^5-t^(15/2))t^(-p)(A+Btκ)E^(-Ctκ),{t,0,Infinity},GenerateConditions->False],n]
Out[]=
-5-p
n
κ
-
19
2
+p
(Cκ)
A
6
C
5
κ
Gamma
7
2
-p+B
5
C
5
κ
Gamma
9
2
-p+
-p
n
κ
-
19
2
+p
(Cκ)
-ACGamma
17
2
-p-BGamma
19
2
-p
In[]:=
-5-p
n
κ
-
19
2
+p
(Cκ)
A
6
C
5
κ
Gamma
7
2
-p+B
5
C
5
κ
Gamma
9
2
-p+
-p
n
κ
-
19
2
+p
(Cκ)
-ACGamma
17
2
-p-BGamma
19
2
-p//.n^(x_):>Zeta[-x-1]/Zeta[-x]
Out[]=
κ
-
19
2
+p
(Cκ)
-ACGamma
17
2
-p-BGamma
19
2
-pZeta[-1+p]
Zeta[p]
+
κ
-
19
2
+p
(Cκ)
A
6
C
5
κ
Gamma
7
2
-p+B
5
C
5
κ
Gamma
9
2
-pZeta[4+p]
Zeta[5+p]
In[]:=
I1=FullSimplify
κ
-
19
2
+p
(Cκ)
-ACGamma
17
2
-p-BGamma
19
2
-pZeta[-1+p]
pZeta[p]
/.{p->17/2+q,κ->1}
Out[]=
2
-1+q
C
(-AC+Bq)Gamma[-q]Zeta
15
2
+q
(17+2q)Zeta
17
2
+q
In[]:=
​
In[]:=
I2=FullSimplify
κ
-
19
2
+p
(Cκ)
A
6
C
5
κ
Gamma
7
2
-p+B
5
C
5
κ
Gamma
9
2
-pZeta[4+p]
pZeta[5+p]
/.{p->19/2-6+q,κ->1}
Out[]=
2
-1+q
C
(AC-Bq)Gamma[-q]Zeta
15
2
+q
(7+2q)Zeta
17
2
+q
In[]:=
​
In[]:=
I1+I2//FullSimplify
Out[]=
20
-1+q
C
(AC-Bq)Gamma[-q]Zeta
15
2
+q
(119+48q+4
2
q
)Zeta
17
2
+q
In[]:=
Factor[(119+48q+4
2
q
)]
Out[]=
(7+2q)(17+2q)
In[]:=
Gamma[-q]Zeta
15
2
+q
(7+2q)(17+2q)Zeta
17
2
+q
/.q->p
Out[]=
Gamma[-p]Zeta
15
2
+p
(7+2p)(17+2p)Zeta
17
2
+p
In[]:=
F==20
-1+q
C
(AC-Bq)
Out[]=
F20
-1+q
C
(AC-Bq)
In[]:=
pts=FunctionPoles
Gamma[-p]Zeta
15
2
+p
(7+2p)(17+2p)Zeta
17
2
+p
,{Abs[p]<50},p
Out[]=
-
97
2
,1,-
93
2
,1,-
89
2
,1,-
85
2
,1,-
81
2
,1,-
77
2
,1,-
73
2
,1,-
69
2
,1,-
65
2
,1,-
61
2
,1,-
57
2
,1,-
53
2
,1,-
49
2
,1,-
45
2
,1,-
41
2
,1,-
37
2
,1,-
33
2
,1,-
29
2
,1,-
25
2
,1,-
21
2
,1,-
17
2
,1,-
13
2
,1,-
7
2
,1,{0,1},{1,1},{2,1},{3,1},{4,1},{5,1},{6,1},{7,1},{8,1},{9,1},{10,1},{11,1},{12,1},{13,1},{14,1},{15,1},{16,1},{17,1},{18,1},{19,1},{20,1},{21,1},{22,1},{23,1},{24,1},{25,1},{26,1},{27,1},{28,1},{29,1},{30,1},{31,1},{32,1},{33,1},{34,1},{35,1},{36,1},{37,1},{38,1},{39,1},{40,1},{41,1},{42,1},{43,1},{44,1},{45,1},{46,1},{47,1},{48,1},{49,1},
1
2
(-17+2ZetaZero[-9]),1,
1
2
(-17+2ZetaZero[-8]),1,
1
2
(-17+2ZetaZero[-7]),1,
1
2
(-17+2ZetaZero[-6]),1,
1
2
(-17+2ZetaZero[-5]),1,
1
2
(-17+2ZetaZero[-4]),1,
1
2
(-17+2ZetaZero[-3]),1,
1
2
(-17+2ZetaZero[-2]),1,
1
2
(-17+2ZetaZero[-1]),1,
1
2
(-17+2ZetaZero[1]),1,
1
2
(-17+2ZetaZero[2]),1,
1
2
(-17+2ZetaZero[3]),1,
1
2
(-17+2ZetaZero[4]),1,
1
2
(-17+2ZetaZero[5]),1,
1
2
(-17+2ZetaZero[6]),1,
1
2
(-17+2ZetaZero[7]),1,
1
2
(-17+2ZetaZero[8]),1,
1
2
(-17+2ZetaZero[9]),1
In[]:=
ComplexListPlot[pts,PlotStyle->{Red},PlotLabel->"Poles of Mellin transform of H[κ]"]
Out[]=
In[]:=
FunctionPoles
Gamma[-p]Zeta
15
2
+p
(7+2p)(17+2p)Zeta
17
2
+p
,p//TableForm
Out[]//TableForm=
-
17
2
1
-
13
2
1
-
7
2
1
1
2
(-17-4

1
) if

1
∈&&

1
≥1
1
2

1
if

1
∈&&

1
≥0
1
1+2

1
if

1
∈&&

1
≥0
1
1
2
(-17+2ZetaZero[

1
]) if

1
∈
Indeterminate
In[]:=
{a,b,c}=ABC[0.3]
Out[]=
{1.14196,0.00305917,0.0601122}
In[]:=
PlotReLog
20c(-ac+bq)Gamma[-q]Zeta
15
2
+q
(119+48q+4
2
q
)Zeta
17
2
+q
(0.1c)^q/.q->0.5+Ix,{x,-5,5},PlotRange->All
Out[]=
In[]:=
Residue
20
-1+q
C
(AC-Bq)Gamma[-q]Zeta
15
2
+q
(119+48q+4
2
q
)Zeta
17
2
+q
,{q,0}
Out[]=
-
20AZeta
15
2

119Zeta
17
2
